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Let $M$ be a smooth manifold.

Let $K$ be a simplicial complex.

Let ${\rm sd}(K)$ be the sub-division of $K$.

Suppose there exists a simplicial sub-complex $K_1$ of ${\rm sd}(K)$ such that $K_1$ is a triangulation of $M$.

Question. Whether or not can we obtain there exists a simplicial sub-complex $K_2$ of $K$ such that $K_2$ is a triangulation of $M$? How about the special case if $M$ is compact and without boundary?

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In general, the answer is no. For an example, suppose that $K$ is the boundary of the four-simplex. Thus $K$ is a triangulation of the three-sphere. Now every closed connected oriented surface of genus $g$, denoted $M_g$, embeds in some subdivision of $K$. However, as $g$ grows, more and more subdivisions are needed. In particular, only the two-sphere, $M_g$, embeds in (the two-skeleton of) $K$.

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    $\begingroup$ It is similar but even a more trivial example is to take $K$ as $2$-simplex. The disjoint union of (at least two) $S^1$s appears as a subcomplex of a subdivision of $K$ but not as a subcomplex of $K$. $\endgroup$
    – Martin Tancer
    Commented May 16 at 9:19

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